Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations

A new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The Bernstein–Petrov–Galerkin method (BPG) is applied to construct the numerical solution for such problems. The method is then tested on examples and compared with other methods. It is shown that the BPG yields better results.